The (coincident) upper bounds for both weak Roman domination and secure domination are provided in Section . Note that \(\gamma _r(J_n) \le \gamma _s(J_n)\) , so a corresponding lower bound for weak Roman domination will also serve as a lower bound for secure domination. We use the formulation for weak Roman domination from [1]} to confirm that \(\gamma _r(J_3) = 5\) , \(\gamma _r(J_4) = 7\) , \(\gamma _r(J_5) = 8\) , \(\gamma _r(J_6) = 10\) , \(\gamma _r(J_7) = 11\) , and \(\gamma _r(J_8) = 13\) , and the formulation from [2]} to confirm the corresponding results for secure domination. Then, suppose there is a value \(k \ge 9\)  such that Theorem  is true for \(n = 3, \hdots , k-1\) , but is not true for \(n = k\) . That is, there exists a weak Roman dominating function \(f\)  for \(J_k\)  with weight \(w(f) < \left\lceil \frac{3k+1}{2} \right\rceil \) . This implies that \(w(f) \le \frac{3k}{2}\) . Hence, from Lemma  we have \(w(f) = \frac{3k}{2}\) , each set of four consecutive copies has weight 6, and each copy has weight 1 or 2.
